Let \(G\)  be a compact quantum group, which means a Hopf \(*\) -algebra \((\mathbb {C}[G],\Delta ,S,\varepsilon )\)  that satisfies the assumption of [1]}. Then, we have the `dual Hopf \(*\) -algebra' \((\mathcal {U}(G),\hat{\Delta },\hat{S},\hat{\varepsilon })\) , whose precise definition is given in the final paragraph of [1]}. Especially, one has to notice that the comultiplication \(\hat{\Delta }\)  is a \(*\) -homomorphism into \(\mathcal {U}(G\times G)\)  rather than \(\mathcal {U}(G)\otimes \mathcal {U}(G)\) , where \(\mathcal {U}(G\times G)\)  is the algebraic dual of the algebraic tensor product \(\mathbb {C}[G]\otimes \mathbb {C}[G]\) . The general principle of Hopf algebra duality suggests that \(\mathcal {U}(G)\)  plays a rôle of group algebra of \(G\) , and the `group \(W^*\) -algebra \(W^*(G)\) ' should be defined inside \(\mathcal {U}(G)\) . A naive intuition is that \(\hat{\Delta }(\mathcal {U}(G)) \subset \mathcal {U}(G\times G)\)  is regarded as a quantum group analogue of \(G < G\times G\) . However, this pair of \(*\) -algebras is too big to discuss its unitary representation theory, and hence we develop the theory in terms of \(W^*(G)\) . We will clarify what \(W^*(G)\)  is by supplying a few facts, because [1]} does not touch it explicitly. The explanation below explicitly or implicitly uses Yamagami's idea [4]}.
